106:, or she can attempt to find a counterexample of the statement if she suspects it to be false. In the latter case, a counterexample would be a rectangle that is not a square, such as a rectangle with two sides of length 5 and two sides of length 7. However, despite having found rectangles that were not squares, all the rectangles she did find had four sides. She then makes the new conjecture "All rectangles have four sides". This is logically weaker than her original conjecture, since every square has four sides, but not every four-sided shape is a square.
114:. For example, suppose that after a while, the mathematician above settled on the new conjecture "All shapes that are rectangles and have four sides of equal length are squares". This conjecture has two parts to the hypothesis: the shape must be 'a rectangle' and must have 'four sides of equal length'. The mathematician then would like to know if she can remove either assumption, and still maintain the truth of her conjecture. This means that she needs to check the truth of the following two statements:
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155:" is the number 2, as it is a prime number but is not an odd number. Neither of the numbers 7 or 10 is a counterexample, as neither of them are enough to contradict the statement. In this example, 2 is in fact the only possible counterexample to the statement, even though that alone is enough to contradict the statement. In a similar manner, the statement "All
272:, counterexamples are usually used to argue that a certain philosophical position is wrong by showing that it does not apply in certain cases. Alternatively, the first philosopher can modify their claim so that the counterexample no longer applies; this is analogous to when a mathematician modifies a conjecture because of a counterexample.
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Callicles might challenge
Socrates' counterexample, arguing perhaps that the common rabble really are better than the nobles, or that even in their large numbers, they still are not stronger. But if Callicles accepts the counterexample, then he must either withdraw his claim, or modify it so that the
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In mathematics, counterexamples are often used to prove the boundaries of possible theorems. By using counterexamples to show that certain conjectures are false, mathematical researchers can then avoid going down blind alleys and learn to modify conjectures to produce provable theorems. It is
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The above example explained — in a simplified way — how a mathematician might weaken her conjecture in the face of counterexamples, but counterexamples can also be used to demonstrate the necessity of certain assumptions and
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counterexample no longer applies. For example, he might modify his claim to refer only to individual persons, requiring him to think of the common people as a collection of individuals rather than as a mob.
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of worse character. Thus
Socrates has proposed a counterexample to Callicles' claim, by looking in an area that Callicles perhaps did not expect — groups of people rather than individual persons.
54:. For example, the fact that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the
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replies that, because of their strength of numbers, the class of common rabble is stronger than the propertied class of nobles, even though the masses are
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As it happens, he modifies his claim to say "wiser" instead of "stronger", arguing that no amount of numerical superiority can make people wiser.
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power. This conjecture was disproved in 1966, with a counterexample involving
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121:"All shapes that have four sides of equal length are squares".
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was disproved by counterexample. It asserted that at least
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a counterexample disproves the generalization, and does so
582:Counterexamples in Probability and Real Analysis
627:Michael Copobianco & John Mulluzzo (1978)
595:Bernard R. Gelbaum, John M. H. Olmsted (2003)
566:Counterexamples in Probability and Statistics
563:Joseph P. Romano and Andrew F. Siegel (1986)
431:Bulletin of the American Mathematical Society
118:"All shapes that are rectangles are squares."
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629:Examples and Counterexamples in Graph Theory
240:Other examples include the disproofs of the
214:imply optimal control laws that are linear.
437:(6). Americal Mathematical Society: 1079.
210:and a linear equation of evolution of the
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75:Suppose that a mathematician is studying
180:powers were necessary to sum to another
98:In this case, she can either attempt to
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147:A counterexample to the statement "all
202:shows that it is not always true (for
580:Gary L. Wise and Eric B. Hall (1993)
569:Chapman & Hall, New York, London
526:Proof in Mathematics: An Introduction
229:is false as shown by counterexamples
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584:. Oxford University Press, New York
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27:Exception to a proposed general rule
616:Second edition, Wiley, Chichester
511:(1976) Cambridge University Press
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102:the truth of the statement using
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196: = 4 counterexamples.
171:Euler's sum of powers conjecture
444:10.1090/s0002-9904-1966-11654-3
613:Counterexamples in Probability
322:Exception that proves the rule
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473:Elkies, Noam (October 1988).
200:Witsenhausen's counterexample
250:Hilbert's fourteenth problem
682:Interpretation (philosophy)
597:Counterexamples in Analysis
549:Counterexamples in Topology
350:"Mathwords: Counterexample"
221:are mappings that preserve
139:Counterexamples in topology
133:Other mathematical examples
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610:Jordan M. Stoyanov (1997)
482:Mathematics of Computation
219:Euclidean plane isometries
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631:, Elsevier North-Holland
399:"What Is Counterexample?"
58:"all students are lazy."
672:Mathematical terminology
524:and Albert Daoud (2011)
56:universal quantification
422:Lander, Parkin (1966).
654:Quotations related to
544:J. Arthur Seebach, Jr.
508:Proofs and Refutations
475:"On A4 + B4 + C4 = D4"
327:Minimal counterexample
188: = 5; other
143:Minimal counterexample
34:is any exception to a
552:, Springer, New York
379:mathworld.wolfram.com
403:www.cut-the-knot.org
248:, the conjecture of
373:Weisstein, Eric W.
206:) that a quadratic
104:deductive reasoning
242:Seifert conjecture
540:Lynn Arthur Steen
534:978-0-646-54509-7
354:www.mathwords.com
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517:0521290384
408:2019-11-28
384:2019-11-28
359:2019-11-28
333:References
270:philosophy
256:, and the
225:, but the
137:See also:
112:hypothesis
89:rectangles
87:that "All
52:philosophy
44:rigorously
453:0273-0979
287:Callicles
165:composite
536:, ch. 6.
458:2 August
311:See also
294:Socrates
227:converse
77:geometry
546:(1978)
282:Gorgias
127:rhombus
93:squares
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161:prime
100:prove
40:logic
633:ISBN
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460:2018
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292:But
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217:All
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